ARTICLE IN PRESS
Journal of Banking & Finance xxx (2010) xxx–xxx
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Journal of Banking & Finance
journal homepage: www.elsevier.com/locate/jbf
Integrated models of capital adequacy – Why banks are undercapitalised
Gavin Kretzschmar a,b,*, Alexander J. McNeil c, Axel Kirchner d
a
University of Edinburgh, Edinburgh, UK
PwC Chair of Accounting and Finance, KIMEP, Almaty, Kazakhstan
c
Maxwell Institute for the Mathematical Sciences, Edinburgh, UK
d
Barrie and Hibbert Limited, Edinburgh, UK
b
a r t i c l e
i n f o
Article history:
Received 26 June 2009
Accepted 26 February 2010
Available online xxxx
JEL classiﬁcation:
G17
G21
G28
Keywords:
Risk management
Economic capital
Enterprise risk management
Basel II
Solvency II
Stochastic models
Stress testing
a b s t r a c t
With the majority of large UK and many US banks collapsing or being forced to raise capital over the
2007–9 period, blaming bankers may be satisfying but is patently insufﬁcient; Basel II and Federal oversight frameworks also deserve criticism. We propose that the current methodological void at the heart of
Basel II, Pillar 2 is ﬁlled with the recommendation that banks develop fully-integrated models for economic capital that relate asset values to fundamental drivers of risk in the economy to capture systematic
effects and inter-asset dependencies in a way that crude correlation assumptions do not. We implement a
fully-integrated risk analysis based on the balance sheet of a composite European bank using an economic-scenario generation model calibrated to conditions at the end of 2007. Our results suggest that
the more modular, correlation-based approaches to economic capital that currently dominate practice
could have led to an undercapitalisation of banks, a result that is clearly of interest given subsequent
events. The introduction of integrated economic-scenario-based models in future can improve capital
adequacy, enhance Pillar 2’s application and rejuvenate the relevance of the Basel regulatory framework.
Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction
The Economic Capital (EC) concept is clear from a technical perspective – it is the capital that a ﬁnancial institution requires in order to operate as a solvent concern at a speciﬁed conﬁdence level
over a given time horizon. In the banking sector, Pillar 2 of Basel
II was speciﬁcally intended to focus on the regulatory review and
internal risk assessment procedures, examining the extent to
which risk management best practices are embedded into bank
decision making. Economic capital modelling and the closely related requirement for stress testing have become fundamental
planks of Pillar 2 compliance (Alexander and Sheedy, 2008). Moreover, banking institutions are required by Pillar 3 to disclose these
risk assessments to external stakeholders. A fundamental problem,
however, is that Pillar 2 EC calculation and Pillar 3 disclosure
requirements exist without clear regulatory guidance as to the
methodology that complex institutional capital models should employ to integrate risk effects across asset classes.
* Corresponding author at: University of Edinburgh, Edinburgh, UK. Tel.: +44 131
650 2448.
E-mail addresses: [email protected] (G. Kretzschmar), a.j.mcneil@
hw.ac.uk (A.J. McNeil), [email protected] (A. Kirchner).
Broadly, EC encapsulates the concept of measuring risk across a
ﬁnancial institution and using the model and its outputs in risk-adjusted comparisons of performance to assist strategic decision
making and deliver value for shareholders. In this paper, we will
argue that the consideration of economic scenarios, their ﬁrmwide effects and the dependencies they induce in asset performances should be the cornerstone of economic capital practice,
and that this requirement should be more clearly articulated in
regulation. In reviewing the current state of ﬁnancial regulation
Brunnermeier et al. (2009) ﬁnd that ‘‘macro-economic analysis
and insight has, in the past, been insufﬁciently applied to the design of ﬁnancial regulation. . .the crisis which began in the US
sub-prime mortgage market in early 2007 and then spread broadly
and deeply was not the ﬁrst banking crisis. It was closer to the
100th. . .”.
A central question concerns the nature of integrated risk methodology used by ﬁnancial institutions for economic capital calculation before and during the current crisis. How were/are risk effects
considered across asset classes and then integrated into a coherent
capital framework? A summary of methodological practice in the
ﬁnancial sector is presented in a comprehensive pre-crisis survey
by the International Financial Risk Institute that included both
banks and insurance companies. In this survey, the prevailing
0378-4266/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.jbankﬁn.2010.02.028
Please cite this article in press as: Kretzschmar, G., et al. Integrated models of capital adequacy – Why banks are undercapitalised. J. Bank Finance (2010),
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G. Kretzschmar et al. / Journal of Banking & Finance xxx (2010) xxx–xxx
approach is reported to be assessment of risk through standalone
models for broad asset classes (or in many cases crude risk categories like market, credit and operational risk) followed by integration using correlation matrices (see IFRI Foundation and CRO
Forum, 2007). This approach to integration was favoured by over
75% of the surveyed banks with the others using simulation approaches or hybrid approaches. In the insurance industry there
was more diversity in the approaches used for integration: around
35% of respondents used the correlation approach and about the
same number used simulation; the remainder reported the use of
copulas or hybrid approaches.
The correlation-based method favoured by so many of the IFRI
respondents, and in particular the banks, is a modular calculation
approach, widely used for its simplicity. In such an approach capital requirements are estimated on a per asset class basis using an
appropriate risk model for that asset class and a risk measure such
as Value-at-Risk (VaR). At the simplest level these per-asset-class
capital requirements can be added although this tends on the
whole (but not always) to overstate capital requirements (Alessandri and Drehmann, 2010; Breuer et al., 2010). Inter-asset diversiﬁcation is typically superimposed using a matrix overlay of
correlation coefﬁcients between asset classes. In this way there is
a resultant downward adjustment to the total capital charge applied to the institution as a whole. A good example of a very detailed application of the modular approach is Rosenberg and
Schuermann (2006), which also shows how copulas can be used
in place of correlations to take better account of dependencies in
the tail.
While modular methods, when carried out carefully, may give
adequate results in ‘‘normal” periods, it has become clear that
the modular approach may prove unreliable in crises and that
the complex interactions of macro-economic factors, ﬁnancial risk
factors, liquidity effects and asset valuations on which economic
capital assessment depends cannot be underpinned by such a simplistic integration approach. Superimposed correlation numbers
are hard to justify, subject to sampling error on account of scarce
data, and, most importantly, make no attempt to tell the narrative
of how correlation arises which is necessary for risk mitigation and
management. In fact, it is essential to understand the sources of
correlation if one wants to measure inter-asset dependencies and
use this to reduce dependencies between different lines of
business.
Integration is an extremely important methodological issue that
requires urgent global regulatory guidance. In a report of the
Financial Stability Forum (2008) supervisors have acknowledged
the need for Pillar 2 principles to strengthen banks’ risk management practices, to sharpen banks’ control of tail risks and to mitigate the build-up of excessive exposures and risk concentrations.
Addressing the methodological deﬁciencies of current treatments
of integration is a major part of this challenge. Our contention in
this paper is that fully integrated factor models based on scenario
generation are the key to addressing this issue. Aggregate risk capital should depend on changes in the valuation of asset positions
which are driven by vectors of risk factors calibrated to real-world
economic conditions. Capital held to support asset positions should
only be reduced by diversiﬁcation due to differences in risk driver
dependencies from position to position. This reﬂects the fact that,
although it may be possible for banks to limit risks by not holding
certain asset classes, it is not possible for bank assets to fully avoid
the pervasive systematic effects of risk factors describing interest
rates, inﬂation, credit, equity and property risk (Alessandri and
Drehmann, 2010; Drehmann et al., 2010).
Although our focus in this paper will be fully-integrated models
at institutional level, this work is taking place against the backdrop
of a wide-ranging review of regulation that raises important ques-
tions about the future of so-called micro-prudential regulation.
Brunnermeier et al. (2009) suggest that regulation has been excessively focussed on seeking to improve the behaviour and risk management practices of individual banks. However, the fullyintegrated approach described in this paper has its counterpart in
integrated models of system-wide risk with additional feedback effects that are being developed by central banks to shed light on
systemic crises and macro-prudential regulation.
The main contributions of this paper are: (i) to demonstrate the
feasibility of fully-integrated economic capital modelling by applying the methodology to a composite balance sheet derived from a
sample of European banks from the pre-crisis period; (ii) to show
how the results suggest a much higher level of capitalisation would
have been desirable than that implied by a typical modular correlation-based approach (i.e. the approach currently used by the
majority of institutions – see IFRI Foundation and CRO Forum
(2007)); (iii) to show how the fully-integrated approach allows
the allocation of this capital to asset classes to gain deeper insights
into the issue of diversiﬁcation. We conclude that there is little surprise that current practice in enterprise risk management failed to
insulate the banking sector against the extreme capital losses that
were incurred.
The paper is structured as follows. In Section 2 we describe the
derivation of an ‘‘average” European bank which will be used for
the empirical investigation of capital adequacy. In Section 3 we
summarise the fully-integrated methodology of the paper, contrast
it with more modular approaches, and describe the architecture of
the economic scenario generation model that we will use. Results
are presented in Section 4 where we devote particular attention
to discussions of fully-integrated projection and fair capital allocation at the institutional level. Section 5 concludes.
2. Construction of an average European bank
To provide empirical insights into the differing effects of implementing both modular and fully-integrated approaches to capital,
we construct a composite 2006 balance sheet of a representative
European bank (EuroBank). Balance sheets for 51 European banks
for the year 2006 are selected to provide a cross-sectional assessment of capital adequacy prior to the credit crisis. Summary statistics presented in the Fifth Quantitative Impact Study, (QIS5, Basel
Committee, 2006) inform our split of aggregate asset positions by
exposure type and credit class, ensuring consistency with asset
proﬁles held by European banks.
The reason we speciﬁcally select European banks as at 2006 is
that Europe offers a fertile ground for investigating the basic effects
of diversiﬁcation on EC in the context of implementing Basel II Pillar 2 regulations. Our data enables a pre ‘‘credit crunch” view of
sector capital adequacy.
We reformulate individual bank balance sheets into a format
that can be utilized to compare EC approaches. Thomson Worldscope database is used to collect an initial sample of 90 banks
whose primary listings are the six largest banking nations in Europe: the United Kingdom (GBR), France (FRA), Germany (GER),
Italy (ITA), Spain (ESP) and the Netherlands (NED).We exclude
small banks (deﬁned as banks with less than £500 million in total
assets), retaining banks which are engaged in at least one of the
following activities: investment banking, deposit-taking or loanmaking. Institutions classiﬁed as Islamic banks are also excluded
as their asset accounting information does not allow the use of
QIS 5 asset mapping characteristics. After exclusions the sample
set is reduced to 51 banks, with the majority of their assets regulated in the UK and the Euro-zone, and therefore subject to Basel II
Pillar 2. Categorisation of individual banks’ balance sheet items
into broader asset classes is informed by notes accompanying the
Please cite this article in press as: Kretzschmar, G., et al. Integrated models of capital adequacy – Why banks are undercapitalised. J. Bank Finance (2010),
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G. Kretzschmar et al. / Journal of Banking & Finance xxx (2010) xxx–xxx
Table 1
Geographic data of the sample set in £ million. This table gives the geographical distribution of the 51 European banks in the sample set (Appendix A, Table A.13). It displays the
total asset value per country, the total number of banks per country and the percentage of total assets in the entire sample set represented by each respective country and in the
ﬁnal column for the entire sample set.
Country
GBR
GER
FRA
ITA
ESP
NED
Europe
Total asset
Number of banks
Percentage (%)
3654142.88
9
36.31
1314256.21
7
13.06
2401069.29
6
23.86
1111829.91
18
11.05
982681.26
9
9.76
600281.90
2
5.96
10064261.47
51
100.00
institutions’ annual reports. Table 1 gives a summary overview of
the distribution of total asset value for the sample. For simpliﬁcation we assume that the composition of EuroBank’s portfolio does
not include proprietary derivative positions, a reasonable assumption given the objective of this work: to compare modular and
integrated EC. Our results are robust to the inclusion of these
derivative positions, since the use of the modular approach is likely
to understate the risk of complex derivatives with non-linear payoff proﬁles. Likewise, risk characteristics of CDOs and RMBS are not
speciﬁcally modelled. The real problem is that disclosures for these
asset classes are often opaque. We classify structured products as
trading book assets and allocate QIS5 type risk characteristics
(note that this conservative treatment strengthens the results of
this work – more capital would be required to support riskier asset
positions).
Credit risky assets are split into ﬁve categories dependening
on their Basel II exposure type: claims on sovereigns, banks, corporates, retail customers and specialized lending. For example,
the capital charge for lending to a corporate is higher than for
lending to a government. As shown in Table 2, lending to corporates and retail/mortgage products are EuroBank’s core business.
To reﬂect credit asset characteristics, we impute the QIS 5 rating
attributes for these classes. Worldscope data disclose nominal
ﬁgures for each bank’s investment and loan portfolio, and so detailed information on the asset composition of each bank’s
investment and loan portfolio is hand collected from the annual
ﬁnancial statements. The majority of banks supply data that enables the derivation of asset composition for investment and loan
portfolios.1 Based on an evaluation of accounting notes contained
in the 2006 Annual Reports we obtain an approximate picture of
weighted average asset holdings in each bank’s investment and
loan portfolios.
The Basel Committee on Banking Supervision has conducted
several Quantitative Impact Studies to gather information to assess
the effect of the Basel II regulatory framework on capital requirements. In the Fifth QIS (Basel Committee, 2006), the probability
distribution of default for every category of credit asset is calibrated and linked to the corresponding credit rating and asset
model (see Crouhy et al. (2000) for full discussion of model alternatives). The percentage of exposure in three PD ranges are
mapped to external credit rating grades of A and better, BBB, and
worse than BBB. Table 3 illustrates the portfolio composition.
We make the simplifying assumption that all sovereign bonds
are AAA rated. For group A and better, we assume that one-third
are AA rated. The category worse than BBB is considered as uniformly BB rated, see Table 4. For example, 38.5% of exposure in corporate loan portfolio show a probability of default less than 0.2%,
1
Off-balance sheet exposures for credit lines are not speciﬁcally modelled. These
are usually representative of on-balance sheet asset characteristics and therefore can
be considered as having a multiplicative scaling effect on the positions we do
consider. Their omission is very unlikely to change our qualitative conclusions with
respect to undercapitalisation. However, this simplifying assumption does not
account for the proportion of undrawn (relative to drawn) credit lines. These could
differ largely across asset portfolios and potentially lead to or reduce concentration
effects.
Table 2
Balance sheet assets (December 2006) of EuroBank in £ million. This table illustrates
the arithmetic average of 51 banks’ balance sheets which will be used as EuroBank’s
balance sheet (last two rows of Appendix A, Table A.13). Derivative positions are
excluded.
Asset class
Average exposure
Cash
Claims on government
Claims on banks
Claims on corporates
Retail loans
ABS
Residential loans
Commercial real estate
Property
Equity
Total assets
% Total assets
2898.33
22716.34
31281.68
65914.72
11441.62
2897.66
37761.59
6061.13
2283.87
14081.54
1
12
16
33
6
1%
19
3
1
7
197338.46
100
Table 3
The calibration of probabilities of default for three categories of credit assets given in
QIS 5 (Basel Committee, 2006). This table displays the calibration of probabilities of
default for three categories of credit assets (bank loan, corporate loan and retail loan)
and three credit ratings (A, BBB and BBB-) given in QIS 5 (Basel Committee, 2006). For
example, 86.2% of claims on banks with PD less than 0.2% are rated as A.
Banks
Corporates
Retails
PD < 0.2% (A)
0.2% 6 PD < 0.8% (BBB)
PD P 0.8%(BBB-)
86.2%
38.5%
30.8%
9.1%
31.8%
34.6%
4.7%
29.7%
34.6%
which implies a credit rating better than BBB for 38.5% of corporate
portfolio. For currency we apply a 70%/30% split to recognise that
balance sheet assets are denominated in both GBP and Eurozone
currency.
The QIS 5 composition parameters for investment and loan
portfolios are applied consistently across all 51 banks to replicate
detailed balance sheet attributes (Appendix A, Table A.13). In projecting and simulating EuroBank, we use Table 2 as the initial balance sheet (arithmetic average of Appendix A, Table A.13). Banks
across the European region differ in size and asset structure; nonetheless the analysis of EuroBank is representative of QIS 5 asset
attributes and is therefore useful for examining the difference between modular and integrated EC calculations.
One very real enterprise risk management (ERM) challenge is
how different portfolios perform over different time horizons. As
one of the key conﬁdence setting parameters in ERM, the time period for capital management directly affects the choice between
conditional (point in time) and unconditional (through the cycle)
calibration processes. Using Table 4, we transform EuroBank’s original balance sheet into a rating based balance sheet (Table 5), and
then compute the EC by the modular approach. For objective comparison with the fully-integrated approach, we use the covariance
matrix proposed by Standard and Poor’s (2008); ﬁrstly, it lacks
bank-speciﬁc institutional bias and secondly, it is an informed
and well-justiﬁed approximation of asset class correlations.
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G. Kretzschmar et al. / Journal of Banking & Finance xxx (2010) xxx–xxx
Table 4
Mapping balance sheet data to asset models using the Fifth Quantitative Impact Study (QIS 5) statistical parameters. This table provides mapping parameters for each asset class
in EuroBank’s balance sheet (Table 2). These percentage parameters are collected from QIS 5 (Basel Committee, 2006) and adjusted by our assumptions. The second and third
columns together represent the asset classes and credit ratings to which balance sheet items are mapped for modelling purposes. For example, 63% and 27% of the claims on
government are mapped into domestic and foreign AAA risk-free nominal bonds, respectively with the remaining 10% mapped to AAA risk-free index-linked bonds.
Modeled as
Credit ratinga
% Split
Dom. 70%
For. 30%
Risk-free nominal bonds
Risk-free index-linked bondsb
Nominal sovereign bonds
Nominal sovereign bonds
Nominal sovereign bonds
Nominal sovereign bonds
AAA
AAA
AA
A
BBB
BB
90%
10%
63%
–
27%
–
Claims on bankd
Nominal corporate bonds
Nominal corporate bonds
Nominal corporate bonds
Nominal corporate bonds
Index-linked corporate bondsb
AA
A
BBB
BB
A
26%
52%
8%
4%
10%
100%
18%
36%
6%
3%
–
8%
16%
2%
1%
–
Claims on corporatese
Nominal corporate bonds
Nominal corporate bonds
Nominal corporate bonds
Nominal corporate bonds
Index-linked corporate bondsb
AA
A
BBB
BB
A
12%
23%
29%
27%
10%
100%
8%
16%
20%
19%
–
3%
7%
9%
8%
–
Retail loansf
Nominal
Nominal
Nominal
Nominal
corporate
corporate
corporate
corporate
bonds
bonds
bonds
bonds
AA
A
BBB
BB
4%
8%
30%
58%
100%
3%
6%
21%
40%
1%
3%
9%
17%
Residential loansg
Nominal
Nominal
Nominal
Nominal
corporate
corporate
corporate
corporate
bonds
bonds
bonds
bonds
AA
A
BBB
BB
4%
8%
30%
58%
100%
3%
6%
21%
40%
1%
3%
9%
17%
Commercial real estate
ABS
Cash
Equities
Property
Nominal corporate bonds
Nominal corporate bonds
Fixed Risk-Free bonds
Equities
Property
BB
BBB
AAA
–
–
100%
100%
100%
100%
100%
70%
70%
–
70%
30%
30%
–
30%
Asset
Claims on government
c
100%
a
For A rated bonds and better, we assume that one-third are AA rated and two-third are A rated.
The proportions for risk-free/corporate Index-linked bonds are all ﬁxed at 10%, the rest (90%) are assigned rating categories according to QIS 5.
c
QIS 5 (Basel Committee, 2006), Table 16 Committee of European Banking Supervisors (CEBS) Group 1 gives a different set of estimates with only 30% AAA, so it may not be
appropriate to use QIS 5 parameters for Sovereigns bonds. We assume that 90% of Sovereign bonds are all AAA rated.
d
QIS 5 (Basel Committee, 2006), Table 15 CEBS Group 1.
e
QIS 5 (Basel Committee, 2006), Table 14 CEBS Group 1.
f
QIS 5 (Basel Committee, 2006), Table 17 Other non-G10 Group 1. QIS 5 does not provide the full PD calibration for Retail but only a simpliﬁed Table 18. The reason we use
Table 17 is that it has similar values of average PD and In Default to Table 18.
g
QIS 5 (Basel Committee, 2006), Table 17 Other non-G10 Group 1. We assume Residential Loans share the same rating parameter with Retail Loans.
b
3. An economic capital modelling framework
3.1. Economic capital and risk measurement
PðDtþ1 EðDt Þ þ Et > 0Þ ¼ a
or equivalently, expressed in terms of losses with Ltþ1 ¼ Dtþ1 , if
PðLtþ1 EðLt Þ < Et Þ ¼ a:
Our economic capital computation for EuroBank will be based
on the application of suitable risk measures to the distribution of
unexpected losses arising from balance sheet positions. These losses
are incurred by value changes in the asset portfolio V t and liabilities Bt due to ﬂuctuations in underlying risk drivers. At the initial
time t, EuroBank is considered to be technically solvent ðV t > Bt Þ
with initial equity value Et ¼ V t Bt . But Et needs to be sufﬁcient
to maintain solvency over the period ½t; t þ 1.
We now take the simplifying assumption that EuroBank replicates their liabilities by a portfolio of assets. We assume that (with
certainty) between time t and t þ 1, the expected increase in asset
value exceeds the increase in the value of liabilities plus any shortfall in income Itþ1 such that
EðV tþ1 V t Þ P ðBtþ1 Bt Þ Itþ1 :
For a given conﬁdence level a (say 99% for a century event) and with
Dtþ1 ¼ V tþ1 V t , the enterprise would be sufﬁciently capitalised if
Now Ltþ1 EðLt Þ is simply the so-called unexpected loss so this
argument justiﬁes setting capital at the a-percentile of the distribution of the unexpected loss.
In general, if we denote the cumulative distribution function of
a generic loss L by F L ðlÞ :¼ PðL 6 lÞ, all risk measures we consider
are statistical measures computed from F L ; in particular we consider Value-at-Risk (VaR), and expected shortfall (ES). The former
is usually deﬁned as the a-quantile of F L for an appropriate choice
of 0 < a < 1, i.e. the measure
VaRa ðLÞ :¼ inffl 2 R : F L ðlÞ P ag;
see McNeil et al. (2005, Deﬁnition 2.10). For economic capital calculation, a is typically chosen to match the target credit rating of the
enterprise (e.g. 99.97% for a AA-rating). The 99.97% VaR is interpreted as indicating that there is a 0.03% chance that the portfolio
loss is at least VaR99:97% .
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G. Kretzschmar et al. / Journal of Banking & Finance xxx (2010) xxx–xxx
Table 5
The transformation of EuroBank balance sheet. This table gives the consolidated balance sheet used in comparing fully-integrated and modular approaches. The fully-integrated
approach models all assets in the ﬁrst column simultaneously with credit-risky assets mapped by credit rating (Table 4). The modular approach models the assets in each of the
last six columns separately, where the corresponding sub-portfolios are mapped to credit ratings individually (Table 4); Economic Capital is then computed using a ﬁxed
correlation matrix. The sovereign sector consists of all cash and claims on government that appear on EuroBank’s balance sheet. The retail sector (ﬁfth column) consists of all retail
loans, residential loans, commercial real estate and asset backed securities (ABS). Derivative positions are not considered.
a
b
Currency £M
Fully integrated
Sovereigns
Institution
Corporate
Retail
Equity
Fixed risk-free AAA
Domestic equities
O’Seas equities
Property
AAA (D)a
AA (D)
A (D)
BBB (D)
BB (D)
AAA (F)b
AA (F)
A (F)
BBB (F)
BB (F)
Index-linked AAA
Index-linked A
Total value
2898.33
9857.08
4224.46
2283.87
14311.29
12438.39
24876.78
27325.32
37306.63
6133.41
5330.74
10661.48
11710.85
15988.56
2271.63
9719.64
197338.46
2898.33
–
–
–
14311.29
–
–
–
–
6133.41
–
–
–
–
2271.63
–
25614.67
–
–
–
–
–
5662.61
11325.22
1793.38
926.25
–
2426.83
4853.66
768.59
396.96
–
3128.17
31281.68
–
–
–
–
–
5329.21
10658.41
13205.35
12333.30
–
2283.95
4567.89
5659.44
5285.70
–
6591.47
65914.72
–
–
4224.46
–
–
1446.57
2893.15
12326.59
24047.08
–
619.96
1239.92
5282.82
10305.89
–
–
58161.98
–
9857.08
–
–
–
–
–
–
–
–
–
–
–
–
–
–
14081.54
Property
–
–
2283.87
–
–
–
–
–
–
–
–
–
–
–
–
2283.87
D is Domestic.
F is Foreign.
Expected shortfall, also used in this paper, is closely related to
the VaR. It is deﬁned as the tail average of the loss distribution
above a given conﬁdence level a. A formal deﬁnition used in Tasche
(2002) and McNeil et al. (2005) is
ESa ðLÞ :¼
1
1a
Z
1
VaRu ðLÞdu;
a
which for continuous loss distributions reduces to the more common expression
ESa ¼
1
EðL1½LP.a ðF L Þ Þ ¼ EðLjL P VaRa Þ;
1a
the expected loss given that the VaR at level a is exceeded.
The question of suitability of a risk measure has been addressed
by Artzner et al. (1999) who propose four axioms which a sound
risk measure should satisfy: monotonicity, subadditivity, positive
homogeneity and translation invariance. Subadditivity implies that
capital charges computed with the risk measure can be reduced by
diversiﬁcation, an important principle in ﬁnance. Conversely, if a
regulator uses a non-subadditive risk measure to determine the
capital charge for a ﬁnancial institution, the institution is incentivised to split its operations into various subsidiaries in an attempt
to reduce the overall capital requirement.
ES, when deﬁned as above, is a coherent risk measure; see
McNeil et al. (2005, Chapter 6). VaR however, is not a coherent risk
measure in general due to non-subadditivity. For comparison, we
compute economic capital requirements under both risk measures
in this paper. In both cases we apply the measures to the distribution of the unexpected loss Ltþ1 EðLtþ1 Þ, which is equivalent to
applying them to Ltþ1 and subtracting the expected loss. Note that
we do not cap losses at 0; ‘negative losses’ are interpreted as gains
but may still lead to positive unexpected losses if gains fall short of
expectations.
3.2. Loss distributions via economic scenario generation
Valuing the portfolio in the present ðV t Þ and in the future ðV tþ1 Þ
is a signiﬁcant challenge that has been recognised by IFRS 7. Current practice favours market-consistent (or fair-value) valuation.
Certain assets are capable of being marked to market while others
are required to be marked to model. We assume that liabilities are
modelled by a matched replicating portfolio of assets. This is a simpliﬁcation we make in order to illustrate the asset valuation differences between modular and integrated methodologies. When
information on liabilities is fully disclosed we would also be able
to model the stochastic ﬂuctuations in liability values.
All asset values at time t can be viewed as being dependent on a
high-dimensional vector of underlying risk factors Zt ¼
ðZ t1 ; . . . ; Z td Þ consisting of such items as equity returns (index and
some single stocks), exchange rates, points on the yield curve,
credit spreads and default or rating migration indicators.
The value of the portfolio at time t can be considered as a random variable of the form
V t ¼ ft ðZt ; tÞ;
ð1Þ
where ft is a function that we will refer to as the portfolio mapping at
time t. It contains information about the portfolio composition at
time t and incorporates the valuation formulas that can be used
to value the more complex (derivative) assets with respect to the
underlying risk factors Zt . Note that, in general, it depends not only
on the value of the risk factors at time t, but also on the time t itself;
this is because the value of a derivative position with maturity/expiry T typically depends on the remaining time to maturity T t.
Note also that there is a time subscript on the mapping function ft
to allow for the possibility of dynamic rebalancing which could
change the entire composition of the mapping over time.
Projecting forward the underlying risk factors for purposes of
valuation at t þ 1 is the role that can be ﬁlled by an economic scenario generator (ESG). We set up a multivariate stochastic process
Z ¼ ðZs ; s P tÞ which projects the values of the risk factors into the
future and gives us snapshots Zs of the economy at future times
s P t. An ESG takes a Monte Carlo (simulation) approach and generates a series of realisations or paths ðZs ðxi Þ; s P tÞ for
i ¼ 1; . . . ; m where each xi is in effect the label for a particular economic scenario.
Risk measures such as VaR and expected shortfall are estimated
by corresponding empirical quantities derived from the Monte Carlo samples, such as sample quantiles. As such, they are prone to
Monte Carlo error, which diminishes with the number of paths
m. Errors and runtimes can be further reduced by employing standard Monte Carlo variance reduction techniques such as the use of
antithetic variates (Robert and Casella, 1999).
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3.3. Implementing the modular and fully-integrated approaches
The asset portfolio of our representative EuroBank may be divided by asset class into d sub-portfolios. For each sub-portfolio
j ¼ 1; . . . ; d we have to consider possible losses
Lj;tþ1 ¼ Dj;tþ1 ¼ ðV j;tþ1 V j;t Þ;
which aggregate by simple summation to give the overall value
change of the enterprise
Ltþ1 ¼ ðV tþ1 V t Þ ¼ d
X
V j;tþ1 j¼1
d
X
j¼1
!
V j;t
¼
d
X
Lj;tþ1 :
j¼1
3.3.1. The modular approach
In the modular approach to capital adequacy individual risks at
sub-portfolio level are transformed into capital charges EC1 ; . . . ;
ECd . These are then combined to calculate the overall economic
capital EC, usually by using a correlation matrix approach.
The economic scenario generation approach gives us the framework for a fully-integrated model of economic capital, but clearly it
also allows us to derive economic capital estimates for individual
asset classes by considering them one at a time. In this way, we
have the opportunity to compare a modular, correlation-based approach to economic capital with a fully-integrated approach. Companies without fully-integrated, enterprise-wide models have no
choice in the matter; they require a method for combining the capital charges that they compute for individual asset classes using a
variety of different models and approaches. The overall EC is generally computed to be
vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u d d
uX X
EC ¼ t
qij EC i EC j ;
i¼1
ð2Þ
j¼1
where qij are the correlations between the asset classes.
The modular method of aggregation is only justiﬁed when
underlying losses in different asset classes have a joint elliptical
distribution and when capital is set using a positive homogeneous,
translation-invariant risk measure, such as VaR or expected shortfall (see McNeil et al., 2005). However, the distributional assumption is hardly ever met in practice and, even if it were, the difﬁculty
of calibrating the correlations and of taking into account tail
dependence, is a serious limitation.
In this paper, we use economic scenario generation to calculate
capital requirements for each asset class using our two risk measures. In other words we set EC i ¼ VaRa ðLi;tþ1 Þ EðLi;tþ1 Þ and
EC i ¼ ESa ðLi;tþ1 Þ EðLi;tþ1 Þ in turn and use (2) to compute overall
economic capital. Standard and Poor’s (2008) also adopt a modular
approach and provide a calibration for the correlation matrix,
which we will use in our analysis. Table 6 shows the correlation
matrix between various credit exposure classes. Standard and
Poor’s (2008) judges that the correlation coefﬁcient between the
credit and equity markets is equal to 80%.
The S&P correlation matrix is part of their ‘‘Risk-adjusted capital
framework for ﬁnancial institutions.” This document appeared in
Table 6
The Standard and Poor’s (2008) correlation matrix For objectivity we use the
correlation matrix used in Standard and Poor’s (2008) own modular approach to
capital calculation. This table gives the correlation coefﬁcients qij which are used in
Eq. (2).
q
Sovereigns
Institutions
Corporates
Retails
Sovereigns
Institutions
Corporates
Retails
100%
75%
50%
25%
–
100%
50%
25%
–
–
100%
25%
–
–
–
100%
April 2008 and summarises the methodology used by S&P to calculate an independent assessment of capital adequacy for ﬁnancial
institutions; the methodology is similar to the Basel II modular
methodology with some adaptations and changes that S&P justify
in the document. Calibration is reported to take a three year perspective, but the correlation matrix has a large element of expert
judgement, as is evident from the round numbers. This matrix is
typical of the kind of correlation matrix used in the modular approach in the pre-crisis period of 2005–07.
We also compute the value of the EC requirement for the case
where there is no diversiﬁcation as a special case of the modular
approach (referred to as simple additive approach) with
P
EC ¼ di¼1 EC i .
In the modular approach, diversiﬁcation could also be measured
by using correlations to calibrate a copula model to join the marginal models together, and to allocate using the composite model.
However, the ‘‘correct” copula will be difﬁcult to obtain and calibrate and we would be sceptical of the value of the results so
obtained.
3.3.2. Fully integrated approach
Losses in sub-portfolios depend on value changes ðLj;tþ1 ¼
Dj;tþ1 ¼ ðV j;tþ1 V j;t ÞÞ and future valuations are driven by fundaðjÞ
ðjÞ
mental risk factors Ztþ1 according to V j;tþ1 ¼ fj;tþ1 ðZtþ1 ; t þ 1Þ. Many
of these risk factors, for example those describing the structure of
the yield curve or the average performance of equity markets, are
common to many sub-portfolios of assets.
This is the origin of dependence in a fully-integrated model:
correlation arises from the mutual dependence of future values
across an enterprise on a set of common risk drivers. Fully inteðjÞ
grated models are common factor models. The risk factors Ztþ1 that
enter into the future valuation of sub-portfolio j contain a subset in
ðkÞ
common with the risk factors Ztþ1 that enter into the future valuation of sub-portfolio k. These common factors are the drivers of
dependence between V j;tþ1 and V k;tþ1 and consequently between
Lj;tþ1 and Lk;tþ1 . The dependence arises endogenously through the
speciﬁcation of the model.
In practical terms we treat the enterprise as a single portfolio
and simulate overall losses for all asset classes and compute capital
using the two risk measures of interest.
3.4. An illustrative example
Suppose we consider a simple balance sheet with three asset
classes: an investment in a stock index, a BBB-rated corporate bond
portfolio; a AAA-rated government bond portfolio. Suppose that
the total portfolio value is 1000 and the initial values of these three
asset classes are 300 for AAA-rated bonds, 600 for BBB-rated bonds
and 100 for equity. Further suppose that each bond portfolio consists of 100 zero-coupon bonds with a common maturity of 10
years.
3.4.1. Model set-up
We assume the equity index St follows a standard geometric
Brownian motion. The valuation of the equity investment is
straightforward, the value function in (1) taking the form
¼ f1 ðSt Þ where f1 is a simple linear scaling function reﬂecting
V equity
t
the size of the investment. The valuation of the bond portfolios in
terms of underlying risk factors is more complicated.
We adopt a ratings-based approach to credit risk in which the
annual default and rating migration probabilities for bonds are
summarised in a matrix P; an element P ij gives the probability of
migrating from rating i to rating j in the course of a year and the
ﬁnal column represents default probabilities. The reduced form
Markov-model approach of Jarrow et al. (1997) (the JLT model) is
used to relate the real-world transition matrix P to a dynamically
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changing set of market-implied default probabilities qit ðTÞ which
can be understood as the market’s implicit assessment of the probability that a bond rated i at time t will default before maturity T.
Following a suggestion of Lando (2004), the JLT model is extended
to incorporate a stochastic credit risk premium process ðpt Þ following a Cox–Ingersoll–Ross (CIR) model; this process can be thought
of as capturing the complex relationship between real-world default rating migration probabilities and credit spreads.
The value of a zero-coupon bond rated i at time t and maturing
at time T is given by pit ðTÞ ¼ pt ðTÞð1 dqit ðTÞÞ where pt ðTÞ is the
price at time t of a default-free zero-coupon bond maturing at T
and d is the loss given default (LGD), which is assumed to be constant. This means that to value a bond portfolio with maturity T at
time t we essentially have a valuation formula of the form
¼ f2 ðpt ðTÞ; pt ; rðtÞÞ where we introduce the vector rðtÞ as a
V bond
t
rating state indicator for all the bonds in the portfolio at time t.
If we know the current price of default-free bonds, the current ratings of the bonds and the value of the credit risk premium process,
we can value the defaultable bonds. To value the default-free bond
we use a 2-factor Black–Karasinski model.
The dependence between equity assets and bond assets is modelled using the popular one-factor approach of Vasicek (1997). Ratings transitions for bond issuer k are considered to be p
driven
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃby
pﬃﬃﬃﬃ
S t þ 1 qkt
latent asset value process of the form Akt ¼ qe
where ðe
S t Þ is a standardised version of the stock index process
above, the ðkt Þ are idiosyncratic noise processes for each bond issuer and q is the parameter known as asset correlation. A series of
deterministic thresholds are created such that the relative value of
Akt with respect to the thresholds dictates the rating of obligor k at
time t. The thresholds are chosen to give the correct matrix of transition probabilities P. In this way the process of rating state variables r(t) is driven by the systematic factor St and the vector of
shock variables t .
and BB
for the shocks effecting the government
Writing AAA
t
t
bond and corporate bond portfolios, respectively, we can summarise the three mapping functions for our asset positions as follows:
V equity
¼ f1 ðSt Þ;
t
bond
V AAA
¼ f2 ðpt ðTÞ; pt ; St ; AAA
Þ;
t
t
bond
V BBB
¼ f2 ðpt ðTÞ; pt ; St ; BBB
Þ:
t
t
Note that the equity index risk factor St is common to the equity
and bond portfolios and induces dependence across all three. The
price of default-free bonds pt ðTÞ and the credit risk premium process pt induce dependence between the different bond portfolios.
3.4.2. Calibration
Calibration of this model involves a number of tasks. First a geometric Brownian motion model for ðSt Þ must be calibrated to historical equity data. We need to choose values for real-world
migration probabilities P using through-the-cycle rating agency
data. We need to calibrate the CIR process for ðpt Þ using data on
corporate bond spreads. The values for the loss-given default d
and the asset correlation q have to be chosen. We also have to calibrate the 2-factor Black–Karasinski interest rate model that will
give pt ðTÞ for any combination of t and T. In addition some correlation between the two main principal components in the equity
model and the Brownian motions that drive the interest rate model
is assumed. More details are available on request.
3.4.3. Results
The results are shown in Table 7. We see that the standalone
economic capitals based on Value-at-Risk at the 99% level for the
three positions are 52 for the AAA government bonds, 109 for
the BBB corporate bonds and 40 for the equity position. Summa-
Table 7
Illustrative example.
Asset
t¼0
t¼1
Exposure
Expected loss
99% VaR loss
99% EC
Standalone
AAA 10y bonds
BBB 10y bonds
Equity
300
600
100
21
46
8
31
63
32
52
109
40
Portfolio
Portfolio loss
–
75
110
185
Capital requirements
EC additive
EC modular
EC fully integrated
–
–
–
–
–
–
–
–
–
201
179
185
qðSovereign=Institutional debt; Corporate debtÞ ¼ 0:5.
qðEquity; BondÞ ¼ 0:8.
tion gives the additive economic capital of 201. On the other hand
use of Standard & Poor’s published correlation numbers and formula (2) gives a reduction in the capital to 179; this is the ﬁgure
we refer to as modular economic capital.
In a fully-integrated economic scenario analysis of the whole
portfolio the computed value for economic capital is 185, which
is slightly larger than the modular ﬁgure. However, it is important
to note that the economic scenario generator has in no way been
calibrated to match the Standard & Poor’s correlations. The ESG
is calibrated at the level of the fundamental interest rate, equity,
credit spread and credit rating migration models as described in
the previous sections.
However, the fully-integrated analysis can be used in a couple
of different ways to inform the choice of modular correlations. It
is possible to search for values of the correlations qij that give
equality between the modular and fully-integrated approaches
(for a ﬁxed VaR level). These could be used and justiﬁed for modular calculations in situations where banks did not have access to
the tools for fully-integrated modelling. Alternatively, the generated data on losses in the three asset classes could be used to estimate a three-dimensional correlation matrix for asset class losses
which could be used in a modular calculation, although there is
no reason why this would match the economic capital number
coming from a fully-integrated analysis, because the three-dimensional loss distribution is not elliptical (see Fig. 1). When we perform this calculation we obtain an economic capital of 174,
which does indeed show the non-elliptical behaviour.
Given this non-elliptical behaviour, we control for possible error
in the estimation of these correlations by using a robust correlation
estimation method based on Kendall’s tau (see McNeil et al., 2005,
pp. 97–98 and 215–217). In fact the correlation numbers are very
similar and the value obtained for economic capital is again 174.
As an aside, in the event of a bank being (partially) funded with
long term liabilities it is possible to ’earn spread risk’ on, say, bonds
issued by the bank itself and which hedge asset side risks. However, the objective of this work is to address the methodological
shortcomings of Basel II and practitioner frameworks by comparing modular vs fully integrated approaches to economic capital.
So, while we certainly accept that a part of EC could be attributed
to spread risk, this would not affect the comparability of modular
and integrated results since both capital computation methods
are applied to a ‘matched asset view’. Under full asset liability
modelling it would certainly be important that the spread effect,
noted above, be accounted for.
3.5. The economic scenario generator
The main study in this paper and the simple example of the previous section are carried out with the Barrie & Hibbert economic
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Fig. 1. Loss distributions for illustrative example (mean corrected).
scenario generator (B&H ESG). Fig. 2 shows the main features of the
model.
The previous section has given some idea of the credit risk modelling approach in the B&H ESG. In this section we give a non-technical overview of further model choices and model calibrations
that have been made to address the economic capital questions
which are of central interest in this paper (See Fig. 2).
3.5.1. Models
Interest rate models are at the core of the ESG and, for the
analyses of this paper, a 2-factor Black–Karasinski model for
nominal interest rates has been used, as its logarithmic structure
guarantees positive nominal rates. Real interest rates are assumed to follow a standard 2-factor Vasicek model, which allows
for positive and negative real rates, while inﬂation is not explic-
Fig. 2. Diagram of model tree used for fully-integrated calculation of EC.
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itly modelled but inferred as the differential between nominal
and real rates.
Since we are limited to ﬁnancial data from annual reports, without full disclosure of the different currencies of assets, we simply
group assets in two economies, domestic and overseas. Exchange
rates between the two economies are modelled based on the
assumption of purchasing power parity (PPP). Over time, real exchange rates are allowed to ﬂuctuate around a long term target;
the deviation of nominal exchange rates from real exchanges rates
is driven by the inﬂation differences between economies.
For equities, we adopt a multi-factor modelling approach where
factors are statistical and derived by principal component analysis
(PCA) from equity index return data. By inversion of the PCA, an
equity index model for the performance of both domestic and
overseas equities is inferred. Property returns are modelled according to the same underlying factor model with appropriate factor
sensitivities derived from empirical analysis.
As indicated in the previous section, the credit risk model in the
ESG combines a Jarrow–Lando–Turnbull (JLT) reduced-form model
ratings-based model with a one-factor Vasicek (or Gaussian copula) model (Jarrow et al., 1997). The extended JLT model allows
for stochastic defaults and migrations of credit-risky assets and is
also able to produce stochastic credit spreads by assuming that
credit risk premia follow a Cox–Ingersoll–Ross (CIR) process.
The ESG modelling suite is used to project the underlying fundamental risk factors for possible future states of the economy.
These are then used to value all balance sheet assets using appropriate valuation models. For simplicity, and because balance sheet
disclosures do not give the information necessary for a more detailed analysis, we treat all credit-risky assets as bond-like assets.
This means that we have the relatively simple task of valuing
risk-free cash ﬂows, equity-like assets (including property) and
defaultable bond-like assets. Thus the approach for the whole balance sheet is in effect a real-world balance sheet expansion of the
example of the previous section.
In the case of the credit-risky assets, these are assumed to be
either nominal or index-linked bonds (see Table 4). The indexlinked coupon bond yields are semi-annually compounded. All
spot rates are continuously compounded.
3.5.2. Calibration
For the calibration process, model parameters of the ESG can
essentially be separated into two sets. Firstly, some ‘‘long term
unconditional targets” are set, based on the statistical analysis of
long series of historical data. These targets generally relate to the
evolution of economic variables and fundamental asset prices such
as volatility, speed of mean reversion and mean reversion level.
These targets are typically updated every quarter or, in some cases,
once per year; they are parameters that are not expected to change
materially from one quarter to the next. Secondly, the remaining
parameters are initialised in a way that is consistent with market
prices where available, at the calibration date. For our study this
calibration date was September 2007, reﬂecting market conditions
at the start of the credit crisis.
Note that the use of long term unconditional targets ensures
long term stability of the suite of models and dampens the effect
of short-term volatility. As a consequence, the model calibrations
for the period prior to the onset of the ﬁnancial crisis, say March
2007, would not be signiﬁcantly different from the ﬁrst ‘‘in crisis”
calibration for September 2007.
The long term targets include variables such as interest rate volatility, equity volatility and equity risk premia, credit spread volatility and long term average levels of spreads, dividend yield
volatility and long terms average levels, the credit rating transition
matrix, exchange rate volatilities and correlations between exchange rates, loss given default and the correlation parameter in
9
Vasicek’s one-factor model of portfolio credit risk. The remaining
‘‘short-term parameters” are calibrated to available market data
at the calibration date; these data include nominal and real yield
curves, equity dividend yields, current spreads and exchange rates.
All market data are based on mid prices.
In addition it is possible to set correlations between the various
Brownian motions that drive the equity, interest rate and other
models. For example, the two main principal components in the
equity model are usually correlated with the Brownian motions
in the interest rate models to better model the observed dependence between these risk factors.
More precise details of calibrated parameter values for some of
the key ESG components which were used in the illustrative example are available on request. The full analysis of the balance sheet
of the average EuroBank of Section 2 uses a larger suite of models
which also includes an exchange rate model (to model the assumed split of assets into domestic and foreign) and a real interest
rate model (to allow the valuation of inﬂation-index-linked bonds).
Full calibration details of these additional models are omitted, but
are available on request.
3.6. EuroBank capital allocation
The advantage of using an integrated risk framework is that
economic capital calculated for an asset portfolio or enterprise
can be broken up into pieces that are attributable to sub-portfolios
or business units. While this is not performed for EuroBank, this
process of capital allocation can be used as the basis of risk-adjusted performance comparison across sub-portfolios and there is
now a considerable literature on the theory of fair allocation of
capital including Tasche (2008), Denault (2001), Kalkbrener
(2005). The generic principle that is commonly adopted is known
as Euler allocation.
Writing L ¼ Ltþ1 ; Li ¼ Li;tþ1 for i ¼ 1; . . . ; d it can be shown that
under some technical assumptions on the distribution of
ðL1 ; . . . ; Ld Þ (fulﬁlled, for example, by the existence of a joint probability density) the Euler contribution to the loss .ðLi jLÞ from asset
class i takes the following forms in the case of VaR and expected
shortfall:
VaRa ðLi jLÞ ¼ EðLi jL ¼ VaRa ðLÞÞ;
ESa ðLi jLÞ ¼ EðLi jL P VaRa ðLÞÞ:
The allocated capital is .ðLi jLÞ EðLi Þ. The forms of these expressions reveal how the economic capital contribution may be estimated using the Monte Carlo output from an economic scenario
generator. For example, in the case of expected shortfall, we would
average the losses in each sub-portfolio over all scenarios where the
total portfolio loss exceeded the Value at risk. In practice, the problem of rare event simulation arises, and long run times may are necessary to get accurate results. But the main point is that the
necessary prerequisite for computing allocations is a fully-speciﬁed
joint model for ðL1 ; . . . ; Ld Þ and this is delivered by a fully integrated
model but not by a modular model and correlation matrix.
A further development, described in Tasche (2006), is the calculation of diversiﬁcation scores to give a measure of the extent of
diversiﬁcation in the total portfolio of an enterprise. A global diversiﬁcation index can be calculated as
.ðLÞ EðLÞ
EC
:
¼ Pd
.
ðL
Þ
EðLÞ
i
i¼1
i¼1 ECi
DI ¼ Pd
This is simply the total economic capital for the portfolio divided by
the sum of standalone economic capital amounts for the subportfolios.
A sub-portfolio diversiﬁcation index can be calculated as
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DIi ¼
.ðLi jLÞ EðLi Þ ACi
:
¼
.ðLi Þ EðLi Þ ECi
This shows the reduction in capital that the sub-portfolio enjoys
through being part of the enterprise. Where the ratio is small, this
is an indication that sub-portfolio i is well-diversiﬁed with respect
to the rest of the enterprise. If the global diversiﬁcation is less
impressive, it may be possible to gain a global improvement by
increasing the size of sub-portfolio i at the expense of other subportfolios.
4. Projecting EuroBank’s balance sheet and computing EC
requirements
4.1. Projected portfolio loss distributions
Following the modular approach, we ﬁrst estimate the loss distribution for sub-portfolios covering credit-risky asset class by
simulation. Descriptive statistics of loss distributions for sub-portfolios of sovereign bonds, interbank lending, corporate bonds and
retail products are shown in the Table 8, describing distributions
for simulated portfolio loss over a one and ﬁve year projection
horizon. Note that every distribution shows a different degree of
skewness. In particular, the loss distribution of retail assets has
the heaviest upper tail out of the four credit risk exposures types.
By contrast, the right hand tail of sovereign bonds is the ‘lightest’.
Results imply that the riskiness of purchasing sovereign bonds is
much lower than mortgage business as would clearly be expected.
We also simulate total returns of equity and property assets over
one and ﬁve years based on an equity multi-factor model to obtain
two corresponding loss distributions. Both loss distributions have
skewed non-normal shapes, which coincides with our expectation
for parent equities, Table 8
4.2. Projected economic capital requirements and capital attribution
under different risk measures
Risk measures VaR and ES for each asset class are computed
based on portfolio loss distributions. Tables 9 and 10 show EC
requirements for every asset class for one-year through ﬁve-year
projections. For the modular approach, we obtain the total EC by
aggregating individual risk measures with and without diversiﬁcation beneﬁt. The special case of aggregation without diversiﬁcation
beneﬁt is referred to as an ‘‘additive” approach where modular EC
is calculated using a correlation matrix overlay. Tables 9 and 10
illustrate EC results under additive and modular approaches with
risk measures 99% VaR, 99% ES and 95% ES. With a fully-integrated
approach, we simulate the total value of the portfolio over one and
ﬁve years and obtain the loss distribution which exhibits a high degree of skewness. EC is computed under three measures of VaR and
ES and given in Tables 9 and 10.
Table 8
Descriptive statistics for loss distributions. This table shows statistical sample parameters for the simulated one-year and ﬁve year loss distributions. All distributions fail the
Jarque–Bera test of normality according to the p-values.
Mean
Median
Panel A: One-year loss distribution
Sovereign
0.09
Institutional
362.28
Corporate
1080.56
Retail
1287.55
Equity
637.16
Property
74.18
Fully integrated
3441.62
Panel A: Five-year loss distribution
Sovereign
62.51
Institutional
2028.07
Corporate
5917.73
Retail
6957.87
Equity
4008.44
Property
463.22
Fully integrated
19168.12
Std. dev.
Skewness
Kurtosis
Max
Min
p-value
24.94
467.17
1246.79
1581.66
309.73
45.45
4030.40
1209.13
1891.23
4478.40
4706.91
2947.55
380.65
13993.46
0.07
0.09
0.11
0.11
0.51
0.46
0.09
0.23
0.03
0.20
0.32
0.31
0.35
0.20
3390.84
5951.95
14669.39
15917.12
6284.98
902.29
45990.98
3572.74
6560.55
15,432.77
16397.23
12062.11
1592.42
48027.38
0.00
0.00
0.00
0.00
0.00
0.00
0.00
61.21
2057.58
5960.25
7067.96
2291.68
288.62
18107.17
2547.93
4077.70
9740.86
10493.26
9361.30
1143.18
31864.71
0.02
0.01
0.02
0.07
1.43
1.06
0.16
0.13
0.01
0.05
0.04
3.57
2.08
0.05
7824.31
10497.51
25339.82
26855.20
12829.63
1844.88
81244.36
8632.50
14795.54
36,061.48
40430.02
59179.59
6525.12
132269.84
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Table 9
One year economic capital requirements and risk measures for the enterprise and individual asset classes. The original balance sheet of EuroBank’s portfolio in €million. The oneyear standalone risk measures for every asset class calculated in modular approach. The one-year Euler risk contributions .ðLi jLÞ EðLi Þ based on three risk measures for every
P
asst class together with their sum di¼1 .ðLi jLÞ EðLi Þ. EC Modular is calculated using a variance–covariance matrix overlay. Figures are in € million.
Asset
Sovereign
Institutional
Corporate
Retail
Equity
Property
Sum
Balance sheet
99% 1 year VaR
t=0
Standalone
Contribution
Standalone
Contribution
25614.67
31281.68
65914.72
58161.98
14081.54
2283.87
2806.85
4129.21
10032.46
9950.04
4946.09
712.10
3214.85
4602.81
10115.11
10312.50
1980.55
192.54
3110.95
4694.06
11448.57
12163.98
5677.10
770.53
2254.84
4480.61
11394.58
12021.01
5305.00
286.01
2523.13
3665.38
8609.56
8970.62
4483.16
617.68
2068.33
3534.87
8550.03
8931.69
3565.85
134.60
197338.46
32576.75
30033.28
37865.19
35742.05
28869.52
26785.36
32576.75
24756.55
–
–
–
30037.00
37865.19
28744.45
–
–
–
35742.06
28869.52
21927.07
–
–
–
26785.38
EC additive
EC modular
EC fully integrated
–
–
–
EC modular (2)
EC modular (3)
–
–
–
–
99% 1 year ES
29580.48
29601.87
–
–
95% 1 year ES
34476.02
34501.01
Standalone
–
–
Contribution
26188.86
26208.24
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Table 10
Five year economic capital requirements and risk measures for the enterprise and individual asset classes. The original balance sheet of EuroBank’s portfolio in €million. The ﬁveyear standalone risk measures for every asset class calculated in modular approach. The ﬁve-year Euler risk contributions .ðLi jLÞ EðLi Þ based on three risk measures for every
Pd
asst class together with their sum i¼1 .ðLi jLÞ EðLi Þ. EC Modular is calculated using a variance–covariance matrix overlay. Figures are in € million.
Asset
Sovereign
Institutional
Corporate
Retail
Equity
Property
Sum
EC additive
EC modular
EC fully integrated
Balance sheet
99% 5 year VaR
t=0
Standalone
Contribution
Standalone
Contribution
Standalone
Contribution
25614.67
31281.68
65914.72
58161.98
14081.54
2283.87
5901.87
7407.95
15600.69
15833.99
10232.94
1401.99
3368.26
5835.90
15048.47
16112.26
6955.03
819.32
6898.74
8614.26
18785.63
18739.28
11131.77
1595.18
5082.56
7714.31
18103.49
18465.21
7434.16
508.80
5356.10
6345.27
13896.11
1397.84
9020.38
1283.42
4290.52
5897.07
13607.96
13696.57
6031.08
-4.69
197338.46
56379.43
48139.25
65764.85
57308.52
49876.12
43518.51
56379.43
43006.76
–
–
–
48926.24
65764.85
50062.16
–
–
–
57041.64
49876.12
38059.94
–
–
–
43763.47
–
–
–
99% 5 year ES
Comparing the EC calculated under all three methods, the additive capital requirements are highest, in line with our expectations
for all three risk measures and both projection horizons (Tables 9
and 10). Under a modular approach, the lowest Economic Capital
number is computed. Modular EC is primarily driven by the correlation matrix used to derive cross asset class diversiﬁcation beneﬁt.
For our study, we used the correlation matrix speciﬁed by Standard
and Poor’s (2008) as a benchmark.
We conclude that the dependence between various assets in
global ﬁnancial markets should be much higher than conventional
assumptions applied in correlation-based calculations. In particular, the equity market and credit market have a strong dependence
on each other, as evidenced by the current credit crisis. Importantly, economic capital calculated over a one year projection horizon shows an undercapitalisation of around 18% for the modular
correlation-based approach compared to the fully-integrated model (Table 9). Modular economic capital remains more than double
the average amount of regulatory capital required under Basel II
Pillar 1 when compared to Bank Tier 1 capital average across 51
banks.
As in the example of Section 3.4, we now calculate the inter-asset-class correlation matrix implied by the fully integrated model
by computing an empirical correlation matrix from the generated
losses. We repeat the modular EC calculation for the one-year time
horizon using this correlation matrix. The results are found in Table 9 in the row marked ‘‘EC modular (2)”. We also use the robust
method of correlation estimation based on Kendall’s tau, as described in Section 3.4; the results for a modular calculation based
on this matrix are shown in the row marked ‘‘EC modular (3)”.
As can be seen, the numbers for the two correlation estimation
methods are quite close, suggesting that the robust method of correlation estimation adds little. In all cases the modular ﬁgures are
lower than the EC ﬁgures coming from a fully-integrated analysis,
revealing the non-elliptical nature of the simulated multivariate
asset loss distribution. However, the discrepancy is much less than
for the S&P correlation matrix. This suggests that output from a
fully integrated analysis can be used to set correlation parameters
for use in a modular calculation, as long as we are aware that there
may still be an inherent tendency to underestimate capital due to
the non-elliptical shape of the underlying loss distribution.
4.3. Capital allocation and diversiﬁcation
As noted earlier, overall capital requirements computed under a
fully integrated approach can be allocated down to sub-portfolios
using Euler Allocation (Tasche, 2008). Euler risk contributions are
calculated for three risk measures and all asset class. The sum of
all attributions is equal to total capital requirement for the fully
95% 5 year ES
integrated approach. Capital requirements attributed to sub-portfolio include respective shares of the total diversiﬁcation beneﬁt
implicit in the fully-integrated EC projection.
The differences between modular sub-portfolio EC calculation
and fully integrated illustrates the diversiﬁcation beneﬁt effect
on a sub-portfolio level and the diversiﬁcation indices of Tasche
(2008) can be computed. Results over one and ﬁve year horizons
for all three risk measures are shown in Tables 11 and 12. From
these table, it is apparent that there is a discount for the contribution to overall capital as measured on a standalone vs contribution
basis. With reference to Table 9, and using 99% ES over 1 year for
example, property requires standalone capital of 770.53 but only
286.01 of capital when it is part of EuroBank’s balance sheet structure. Naturally this computation is sensitive to model choice and
Table 11
One year diversiﬁcation indices. The one-year marginal DI. ðLi jLÞ of every asset class
with respect to three risk measures. The one-year ‘‘absolute” DI. ðLÞ of whole portfolio
with respect to three risk measures.
Asset
Risk measure
99% ES
95% ES
Panel A: Marginal diversiﬁcation index
Sovereigns
1.145
Institutions
1.115
Corporates
1.008
Retails
1.036
Equity
0.400
Property
0.270
99% VaR
0.725
0.955
0.995
0.988
0.934
0.371
0.820
0.964
0.993
0.996
0.795
0.218
Panel B: Absolute diversiﬁcation index
Total portfolio
0.922
0.944
0.928
Table 12
Five year diversiﬁcation indices. The ﬁve-year marginal DI. ðLi jLÞ of every asset class
with respect to three risk measures. The ﬁve-year ‘‘absolute” DI. ðLÞ of whole portfolio
with respect to three risk measures.
Asset
Risk measure
99% ES
95% ES
Panel A: Marginal diversiﬁcation index
Sovereigns
0.195
Institutions
0.473
Corporates
0.794
Retails
0.917
Equity
1.233
Property
0.778
99% VaR
0.737
0.896
0.964
0.985
0.668
0.319
0.801
0.929
0.979
0.980
0.669
-0.004
Panel B: Absolute diversiﬁcation index
Total portfolio
0.867
0.867
0.877
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G. Kretzschmar et al. / Journal of Banking & Finance xxx (2010) xxx–xxx
calibration, but unlike the covariance approach it is possible to isolate the source and cause of the capital diversiﬁcation.
4.4. Summary: A framework for management and regulatory action
In the modular approach, correlations are somewhat arbitrary
and hard to justify. The fully-integrated approach gives a more
structural and explanatory way to construct dependence of assets
on risk factors for which data and policies are capable of being analysed. The fully integrated approach enables a risk-based allocation
of capital and facilitates ‘‘use” by permitting the isolation of worst
case paths for EC. In this way, it provides a clear framework for
informing management actions. The main points taken from our
results suggest that:
Fully integrated capital is greater than modular capital but less
than additive capital. See Table 9.
The main contributions to fully-integrated capital come from
corporate lending and retail advances; this is a function of the
effect of risk factors on balance sheet exposures to assets-and
the credit risk rating embodied in our credit risk calibration.
See Table 9 (column 2) for capital contributions.
The overall diversiﬁcation score is high; see Table 11. This score
should be taken as measuring diversiﬁcation potential rather
than absolute diversiﬁcation. In other words the overall balance
sheet is quite concentrated and there is potential to improve
diversiﬁcation by moving into asset classes that have lower
diversiﬁcation scores (and which are thus better diversiﬁed).
However, we note that the dependence assumptions are quite
Table A.13
Balance sheet assets (December 2006) of 51 European banks in £ million. This table displays 51 European banks’ balance sheets, modiﬁed using detailed accounting notes. The last
three rows summarise the total value of the balance sheets, the average which we will use as the EuroBank’s balance sheet and the percentages of the composition. This table does
not take account of derivative positions.
Ticker
Country
Cash
Claims
on gov.
Claims on
banks
Claims
on corp.
Retail
loans
ABS
Res. loans
Comm.
real estate
Prop.
Eq.
Total
assets
AABA-AE
ACA-FR
AL.-LN
BARC-LN
BB.-LN
BBVA-MC
BDB-MI
BEB2-FF
BKT-MC
BMPS-MI
BNP-FR
BPE-MI
BPI-MI
BPSO-MI
BPVN-MI
BTO-MC
BVA-MC
CAP-MI
CBK-FF
CC-FR
CE-MI
CRG-MI
CVAL-MI
DBK-FF
DPB-FF
EHY-FF
GLE-FR
GUI-MC
HBOS-LN
HSBA-LN
IKB-FF
ISP-MI
KN-FR
LANS-AE
LLOY-LN
MB-MI
MEL-MI
NRK-LN
OLB-FF
PAS-MC
PEL-MI
PIN-MI
PMI-MI
POP-MC
RBS-LN
SAB-MC
SAN-MC
STAN-LN
TRNO-FR
UBI-MI
UC-MI
Total
Average
%
NED
FRA
GBR
GBR
GBR
ESP
ITA
GER
ESP
ITA
FRA
ITA
ITA
ITA
ITA
ESP
ESP
ITA
GER
FRA
ITA
ITA
ITA
GER
GER
GER
FRA
ESP
GBR
GBR
GER
ITA
FRA
NED
GBR
ITA
ITA
GBR
GER
ESP
ITA
ITA
ITA
ESP
GBR
ESP
ESP
GBR
FRA
ITA
ITA
8299
10,637
2390
9753
233
8432
70
643
367
1214
7983
482
410
47
781
282
83
1076
3456
6208
372
395
226
4722
684
83
6305
115
2846
22,068
33
3645
513
35
3329
46
14
956
43
570
30
31
332
1012
6121
610
13,734
3934
18
273
11,876
147,815
2898.33
1%
52,416
104,292
3308
152,788
1285
17,256
379
9171
1951
5862
136,077
1152
1161
503
1688
5439
129
3241
35,778
19,986
1147
896
272
123,463
12,066
9738
91,368
120
40,670
78,148
2387
11,322
48,165
229
21,510
2695
99
1670
151
240
178
78
1250
1191
61,181
1083
37,970
9317
16
2089
43,965
1,158,533
22716.34
12%
92,955
112,107
10,177
137,133
7138
38,969
702
13,663
4593
13,876
127,131
4473
4030
1543
6791
10,361
1662
13,104
59,916
18,740
2339
2296
1441
90,187
18,031
23,124
85,510
929
86,156
131,680
4617
28,495
38,662
1950
49,392
4465
379
15,968
945
2400
740
413
3891
9436
123,944
7060
78,002
18,484
238
7212
77,916
1,595,366
31281.68
16%
201,340
235,869
21,310
267,327
15,120
83,161
1511
28,209
9465
33,284
263,670
9610
9864
3212
14,698
21,640
3563
29,128
122,667
36,038
4911
5052
3144
200,227
36,991
47,808
174,873
1982
181,455
281,316
9486
60,369
81,407
4247
105,285
9110
810
34,112
2015
5166
1554
900
8479
20,390
266,102
16,002
167,329
41,136
508
15,069
163,729
3,361,651
65914.72
33%
39,267
33,014
5148
31,579
3973
18,109
302
5260
2164
6547
31,220
2368
2091
780
3616
4518
985
6984
24,615
4627
1048
1109
798
12,744
6954
10,921
21,054
533
39,158
54,266
2026
13,695
7126
1128
23,080
1824
200
9306
533
1403
396
228
1973
5428
55,400
3994
35,051
8194
142
3737
32,904
583,522
11441.62
6%
6686
13,303
422
19,489
164
2201
48
1170
249
748
17,358
147
148
64
215
694
16
413
4564
2549
146
114
35
15,749
1539
1242
11,655
15
5188
9968
305
1444
6144
29
2744
344
13
213
19
31
23
10
159
152
7804
138
4843
1188
2
266
5608
147,781
2897.66
1%
129,597
108,960
16,990
104,223
13,111
59,768
997
17,359
7141
21,608
103,036
7816
6899
2574
11,936
14,910
3250
23,050
81,239
15,272
3460
3660
2633
42,058
22,952
36,045
69,487
1760
129,237
179,099
6686
45,198
23,517
3722
76,172
6021
661
30,714
1758
4629
1308
754
6513
17,915
182,840
13,183
115,683
27,044
467
12,334
108,597
1,925,841
37761.59
19%
20,802
17,489
2727
16,729
2104
9593
160
2786
1146
3468
16,538
1255
1107
413
1916
2393
522
3700
13,040
2451
555
588
423
6751
3684
5786
11,153
283
20,744
28,747
1073
7255
3775
597
12,226
966
106
4930
282
743
210
121
1045
2875
29,348
2116
18,568
4341
75
1980
17,431
309,117
6061.13
3%
4224
4650
556
2492
91
3175
102
469
232
1728
12,318
625
638
87
363
634
157
1959
935
939
227
795
313
2795
684
209
7373
84
11,264
8393
161
1973
1145
126
8991
209
3
197
74
229
114
44
502
477
18,420
662
6812
1108
10
908
5805
116,477
2283.87
1%
32,492
64,649
2051
94,711
796
10,697
235
5685
1209
3634
84,352
714
720
312
1046
3371
80
2009
22,178
12,389
711
555
168
76,533
7480
6036
56,638
75
25,211
48,443
1480
7019
29,857
142
13,333
1671
61
1035
93
149
110
49
775
738
37,925
671
23,537
5775
10
1295
27,253
718,158
14081.54
7%
588,077
704,971
65,077
836,224
44,015
251,362
4507
84,415
28,516
91,968
799,684
28,642
27,069
9535
43,050
64,241
10,445
84,663
368,387
119,200
14,917
15,460
9452
575,229
111,065
140,992
535,417
5896
541,928
842,129
28,255
180,414
240,311
12,204
316,062
27,352
2347
99,101
5913
15,558
4662
2628
24,918
59,614
789,085
45,519
501,529
120,521
1486
45,163
495,083
10,064,261
197338.46
100%
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conservative in the model. We would expect that the inclusion
of complex assets and derivative positions might enhance the
picture of diversiﬁcation.
Corporate and retail have high diversiﬁcation indexes (see Table
11) whereas equity and property are lower, reﬂecting potential
to diversify by moving more assets to the latter classes.
Conclusions are broadly similar for all risk measures: additive
capital is the highest with modular the lowest; fully-integrated
EC consistently falls between the two.
5. Conclusion
At an institutional level we observe materially different results
for Economic Capital computations for identical asset classes under
modular and fully-integrated approaches. Both are methods currently permissible under Basel II, Pillar 2. The modular approach
uses a correlation matrix overlay to capture dependence between
different asset class risks. By contrast, in the fully-integrated approach, correlations are due to mutual dependencies in the driving
risk factors in global markets. The comparison of the two approaches shows that, precisely in stress episodes, such as credit
contagion, capital derived using a correlation matrix is discrepant
with the fully-integrated framework (and can only be accordant
by accident). In summary the fully-integrated approach:
Avoids theoretical pitfalls and practical limitations of more
modular approaches.
Opens the door to capital allocation, risk-adjusted performance
comparison and risk-based enterprise steering, as is the ultimate goal of Enterprise Risk Management.
Provides a framework for rational (probability-based) stress
testing. It is possible to identify the risk factors that ‘‘correlate”
highly with asset value losses and reveal the factors that are
particularly inﬂuential in the tail, i.e. we can get a proper handle
on tail dependence.
Allows the isolation of model and calibration effects on EC.
Provides capital results that can allow the consideration of path
dependent actions, such as portfolio rebalancing.
Finally, it is clear that different risk measures and different approaches give different risk capital ﬁgures, and the ‘‘undercapitalisation” implied by a modular approach with respect to a fullyintegrated approach, while interesting in the current climate, is
not really the main message of this paper. After all, it might be argued that this undercapitalisation can be simply rectiﬁed by
increasing the correlations. We believe that this is the wrong conclusion and that risk management can never be about the manipulation of poorly understood numbers to obtain the most
convenient set of results.
Rather the main message of this paper is that the fully-integrated scenario-based approach offers a powerful explanatory
framework for integrated risk management. A regulatory emphasis
on the development of such models for calculating economic capital would greatly enhance Pillar 2’s application and indeed rejuvenate the relevance of Basel II. A focus on the economic drivers of
risk and their systematic (and systemic) effects would be expected
to lead to better capitalisation standards in future.
13
Acknowledgements
Opinions expressed here are our own and do not necessarily reﬂect the views of Barrie & Hibbert Limited. We are grateful for useful discussions with the Bank of England ﬁnancial stability
modelling team, Standard & Poor’s and Cass Business Forum participants at the Money Macro and Finance group conference, and participants at the INFINITI 2009 Dublin conference. Insightful
comments from our discussant, Aneta Hryckiewicz, at the latter
conference and from an anonymous referee have greatly contributed to the reﬁnement of this work. We acknowledge the research
assistance of Yuan Zhou and Liliya Sharifzyanova.
Appendix A. Balance sheet assets of 51 European banks
See Table A.13.
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Please cite this article in press as: Kretzschmar, G., et al. Integrated models of capital adequacy – Why banks are undercapitalised. J. Bank Finance (2010),
doi:10.1016/j.jbankﬁn.2010.02.028